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What is special about Maxwell's equations? If I have read correctly, what Maxwell basically did is combine 4 equations that were already formulated by other physicists as a set of equations. Why are these 4 equations (out of large numbers of mathematical equations in electromagnetism) important? Or What is special about these 4 equations?

Qmechanic
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Four Seasons
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    http://galileoandeinstein.physics.virginia.edu/more_stuff/Maxwell_Eq.html –  Mar 31 '17 at 04:35
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    What about reading e.g. the introduction to the Wikipedia article did not already answer this in a broad sense? Please show research effort when asking questions and be more specific about what you want to know. – ACuriousMind Mar 31 '17 at 11:47
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    "what Maxwell basically did is combine 4 equations that were already formulated by other physicists" Well, no. He extended and corrected one of the existing equations, first. – dmckee --- ex-moderator kitten Mar 31 '17 at 15:49
  • Is the question why we care about these equations today (the importance of the equations themselves, without caring much about who did what when), or a historical perspective (what did Maxwell specifically do, relative to his peers and predecessors)? One can be more “special” than the other. (In linguistics these are called the synchronic and diachronic perspectives.) – ShreevatsaR Mar 31 '17 at 18:02
  • @ShreevatsaR In fact, the questions can be answered from both perspectives. – Four Seasons Apr 03 '17 at 05:52

3 Answers3

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Maxwell's equations wholly define the evolution of the electromagnetic field. So, given a full specification of an electromagnetic system's boundary conditions and constitutive relationships (i.e. the data defining the materials within the system by specifying the relationships between the electric / magnetic field and electric displacement / magnetic induction), they let us calculate the electromagnetic field at all points within the system at any time. Experimentally, we observe that knowledge of the electromagnetic field together with the Lorentz force law is all one needs to know to fully understand how electric charge and magnetic dipoles (e.g. precession of a neutron) will react to the World around it. That is, Maxwell's equations + boundary conditions + constitutive relations tell us everything that can be experimentally measured about electromagnetic effects (including quibbles about the Aharonov-Bohm effect, see 1). Furthermore, Maxwell's equations are pretty much a minimal set of equations that let us access this knowledge given boundary conditions and material data, although, for example, most of the Gauss laws are contained in the other two given the continuity equations. For example, if one takes the divergence of both sides of the Ampère law and applies the charge continuity equation $\nabla\cdot\vec{J}+\partial_t\,\rho=0$ together with an assumption to $C^2$ (continuous second derivative) fields, one derives the time derivative of the Gauss electric law. Likewise, the divergence of the Faraday law yields the time derivative of the Gauss magnetic law.

Maxwell's equations are also Lorentz invariant, and were the first physical laws that were noticed to be so. They're pretty much the simplest linear differential equations that possibly could define the electromagnetic field and be generally covariant; in the exterior calculus we can write them as $\mathrm{d}\,F = 0;\;\mathrm{d}^\star F = \mathcal{Z}_0\,^\star J$; the first simply asserts that the Faraday tensor (a covariant grouping of the $\vec{E}$ and $\vec{H}$ fields) can be represented as the exterior derivative $F=\mathrm{d} A$ of a potential one-form $A$ and the second simply says that the tensor depends in a first order linear way on the sources of the field, namely the four current $J$. This is simply a variation on Feynman's argument that the simplest differential equations are linear relationships between the curl, divergence and time derivatives of a field on the one hand and the sources on the other (I believe he makes this argument in volume 2 of his lecture series, but I can't quite find it at the moment).

1) Sometimes people quibble about what fields define experimental results and point out that the Aharonov-Bohm effect is defined by the closed path integral of the vector magnetic potential $\oint\vec{A}\cdot\mathrm{d}\vec{r}$ and thus ascribe an experimental reality to $\vec{A}$. However, this path integral of course is equal to the flux of $\vec{B}$ through the closed path, therefore knowledge of $\vec{B}$ everywhere will give us the correct Aharonov-Bohm phase for to calculate the electron interference pattern, even if it is a little weird that $\vec{B}$ can be very small on the path itself.
AccidentalFourierTransform
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Selene Routley
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    I'm tempted to say that Maxwell's equations are important because they let us know that Lorentz invariance is a thing; the fact that they also unified electromagnetism is a handy side-effect. ;) – PM 2Ring Apr 01 '17 at 09:11
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    @PM2Ring True; that was their role historically, but one can equally well derive the Lorentz transformation from basic symmetry assumptions about the Universe quite independently of light and Maxwell's equations, as I discuss here. Experimentally, one observes the speed of light to have the same invariance as the $c$ that drops out of this other approach. One can then go on to derive Maxwell's equations from relativity a number of ways; one of the funnest is simply to assume that the effect of the Coulomb inverse square law for charges ... – Selene Routley Apr 01 '17 at 13:48
  • ...@PM2Ring propagates at $c$ as required (or at least, as limited) by relativity; you can then derive Maxwell's equations as a Lorentz covariant description of this idea. – Selene Routley Apr 01 '17 at 13:49
  • Re Aharonov-Bohm: but what if spacetime isn't simply connected?! – Holographer Apr 02 '17 at 14:11
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Maxwell's equations embedded fundamental laws for electricity and magnetism. All empirical formulations fitting the measurements at that time, were tied elegantly into a unified mathematical theory of electromagnetism, which fitted the data and was very predictive.

It was from Maxwell's equations that electromagnetic waves were predicted to exist and described mathematically light and other EM radiations.

In this sense Maxwell's equations are fundamental because of the unification of the electric field and magnetic field formalism into one mathematical model, and for the discovery of the equations ruling light/radiation.

(It was the basic role model for the unification of the electromagnetic and weak interactions in particle physics, and the models unifying strong-weak-electromagnetic.)

anna v
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Because they are fundamental, and Maxwell was able to take a subject as broad as electromagnetism and form an accurate description of virtually everything in it, using only 4 equations that give rise to other, less general equations.

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    Actually, Maxwell wrote down like 20 equations. It was Oliver Heaviside who reduced this to 4 equations using vector calculus. –  Mar 31 '17 at 20:40